Pinning phenomena in the Ginzburg-Landau Model of Superconductivity

We study the Ginzburg-Landau energy of superconductors with a term $a_\ep$ modelling the pinning of vortices by impurities in the limit of a large Ginzburg-Landau parameter $\kappa=1/\ep$. The function $a_\ep$ is oscillating between 1/2 and 1 with a scale which may tend to 0 as $\kappa$ tends to infinity. Our aim is to understand that in the large $\kappa$ limit, stable configurations should correspond to vortices pinned at the minimum of $a_\ep$ and to derive the limiting homogenized free-boundary problem which arises for the magnetic field in replacement of the London equation. The method and techniques that we use are inspired from those of Sandier-Serfaty (in which the case $a_\ep \equiv 1$ was treated) and based on energy estimates, convergence of measures and construction of approximate solutions. Because of the term $a_\ep(x)$ in the equations, we also need homogenization theory to describe the fact that the impurities, hence the vortices, form a homogenized medium in the material.